VSRP, BAYESCOMP @ CEMSE-KAUST
https://alvinbjl.github.io/school-disparity-areal-model/presentation/
October 8, 2025
Education is a foundational pillar of national development and its people. Key factors for quality education:
Education is a foundational pillar of national development and its people. Key factors for quality education:
Several studies examined general aspects of education in Brunei [Ebil and Shahrill (2023); Abdul Latif, Matzin, and Escoto-Kemp (2021); Salbrina, Deterding, and Nur Raihan (2024); Mohamad et al. (2018).
First to examine through spatial methods & statistical models!
Main goal
Identify adminstrative regions in Brunei where school availability falls significantly below the national baseline.
This supports future policy planning and school placements.
Study Region: Brunei Darussalam
School Specification: Primary & Secondary Government (PUBLIC) schools
Region: 39 Mukim (sub-division of district)
Bruneiverse GitHub page mainly via bruneimap R package (Jamil 2025; Jamil et al. 2025)Let \(Y_i\) and \(E_i\) denote the observed and expected counts of schools, respectively, in mukim \(i \in \{1, \dotsc, n\}\). Let \(\theta_i\) represent the relative abundance (RA) of schools in mukim \(i\), analogous to a relative risk in disease mapping.
Bayesian hierarchical model + BYM
\[ Y_i \mid \theta_i \sim \text{Poisson}(E_i \cdot \theta_i), \quad i = 1, \dotsc, n \]
\[ \log(\theta_i) = \beta_0 + \beta_1 \cdot \text{pop}_i + \beta_2 \cdot \text{area}_i + \beta_3 \cdot \text{hp}_i + u_i + v_i, \]
Bayesian hierarchical model + BYM
\[ Y_i \mid \theta_i \sim \text{Poisson}(E_i \cdot \theta_i), \quad i = 1, \dotsc, n \]
\[ \log(\theta_i) = \beta_0 + \beta_1 \cdot \text{pop}_i + \beta_2 \cdot \text{area}_i + \beta_3 \cdot \text{hp}_i + u_i + v_i, \]
Spatial random effect \(u_i\) uses Queen Contiguity neighborhood (adjacency)
Model fitting performed using INLA (Rue, Martino, and Chopin 2009).
Model adequacy (spatial structure) is assessed by spatial autocorrelation (Global Moran’s I) of residual: \[residual_i = \dfrac{Y_i - \mu_i}{\sqrt{\mu_i}} = \dfrac{Y_i - E_i \cdot \theta_i}{\sqrt{E_i \cdot \theta_i}},\]
For each mukim in the study area \(i, j = 1, 2, \ldots, N\). The Moran’s I test statistic is defined as follows:
\[ I = \frac{N}{\sum_{i=1}^N \sum_{j=1}^N w_{ij}} \frac{\sum_{i=1}^N \sum_{j=1}^N w_{ij} (x_i - \bar{x})(x_j - \bar{x})}{\sum_{i=1}^N (x_i - \bar{x})^2} \in [-1,1], \]
Same Queen Contiguity neighbour is used.
Central Limit Theorem (CLT) employed for hypthesis test for significance of the Moran’s I statistic (z-scores):
| Term | Mean | 2.5% | 97.5% |
|---|---|---|---|
| (Intercept) | 1.076 | 0.239 | 1.893 |
| pop_s | -0.436 | -0.579 | -0.295 |
| area_s | 0.015 | -0.002 | 0.032 |
| hp_s | -0.759 | -3.174 | 1.705 |
